"A good stock of examples, as large as possible, is indispensable for a thorough understanding of any concept, and when I want to learn something new, I make it my first job to build one." – Paul Halmos

Archive for April, 2013

A common theme in mathematics is to replace the study of an object with the study of some category that can be built from that object. For example, we can

replace the study of a group with the study of its category of linear representations,

replace the study of a ring with the study of its category of -modules,

replace the study of a topological space with the study of its category of sheaves,

and so forth. A general question to ask about this setup is whether or to what extent we can recover the original object from the category. For example, if is a finite group, then as a category, the only data that can be recovered from is the number of conjugacy classes of , which is not much information about . We get considerably more data if we also have the monoidal structure on , which gives us the character table of (but contains a little more data than that, e.g. in the associators), but this is still not a complete invariant of . It turns out that to recover we need the symmetric monoidal structure on ; this is a simple form of Tannaka reconstruction.